Homology and cell structure of nilpotent spaces
Author:
Robert H. Lewis
Journal:
Trans. Amer. Math. Soc. 290 (1985), 747760
MSC:
Primary 55P99; Secondary 20C07, 57M99
MathSciNet review:
792825
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let and denote finitely dominated nilpotent complexes. We are interested in questions relating the homology groups of such spaces to their cell structure and homotopy type. We solve a problem posed by Brown and Kahn, that of constructing nilpotent complexes of minimal dimension. When the fundamental group is finite, the threedimensional complex we construct may not be finite; we then construct a finite sixdimensional complex. We investigate the set of possible cofibers of maps , and find a severe restriction. When it is met and the fundamental group is finite, can be constructed from by attaching cells in a natural way. The restriction implies that the classical notion of homology decomposition has no application to nilpotent complexes. We show that the Euler characteristic of must be zero. Several corollaries are derived to the theory of finitely dominated nilpotent complexes. Several of these results depend upon a purely algebraic theorem that we prove concerning the vanishing of homology of nilpotent modules over nilpotent groups.
 [1]
Kenneth
S. Brown, Euler characteristics of discrete groups and
𝐺spaces, Invent. Math. 27 (1974),
229–264. MR 0385007
(52 #5877)
 [2]
Kenneth
S. Brown and Peter
J. Kahn, Homotopy dimension and simple cohomological dimension of
spaces, Comment. Math. Helv. 52 (1977), no. 1,
111–127. MR 0438336
(55 #11251)
 [3]
Gunnar
Carlsson, A counterexample to a conjecture of Steenrod,
Invent. Math. 64 (1981), no. 1, 171–174. MR 621775
(82j:57036), http://dx.doi.org/10.1007/BF01393939
 [4]
William
G. Dwyer, Vanishing homology over nilpotent
groups, Proc. Amer. Math. Soc. 49 (1975), 8–12. MR 0374242
(51 #10442), http://dx.doi.org/10.1090/S00029939197503742423
 [5]
Karel
Ehrlich, The obstruction to the finiteness of the total space of a
fibration, Michigan Math. J. 28 (1981), no. 1,
19–38. MR
600412 (82i:55014)
 [6]
K.
Hoechsmann, P.
Roquette, and H.
Zassenhaus, A cohomological characterization of finite nilpotent
groups, Arch. Math. (Basel) 19 (1968), 225–244.
MR
0227278 (37 #2863)
 [7]
P. J. Hilton, Homotopy theory and duality, Gordon & Breach, New York, 1966.
 [8]
Robert
H. Lewis, Equivariant cofibrations and
nilpotency, Trans. Amer. Math. Soc.
267 (1981), no. 1,
139–155. MR
621979 (82j:55009), http://dx.doi.org/10.1090/S00029947198106219791
 [9]
, Ph. D. dissertation, Cornell University, 1977.
 [10]
Guido
Mislin, Wall’s obstruction for nilpotent spaces,
Topology 14 (1975), no. 4, 311–317. MR 0394647
(52 #15448)
 [11]
Guido
Mislin, Finitely dominated nilpotent spaces, Ann. of Math. (2)
103 (1976), no. 3, 547–556. MR 0415607
(54 #3690)
 [12]
G.
Mislin, Groups with cyclic Sylow subgroups and finiteness
conditions for certain complexes, Comment. Math. Helv.
52 (1977), no. 3, 373–391. MR 0448344
(56 #6651)
 [13]
Guido
Mislin, The geometric realization of Wall obstructions by nilpotent
and simple spaces, Math. Proc. Cambridge Philos. Soc.
87 (1980), no. 2, 199–206. MR 553576
(81f:55011), http://dx.doi.org/10.1017/S0305004100056656
 [14]
G.
Mislin and K.
Varadarajan, The finiteness obstructions for nilpotent spaces lie
in 𝐷(𝑍𝜋), Invent. Math. 53
(1979), no. 2, 185–191. MR 560413
(81e:55011), http://dx.doi.org/10.1007/BF01390032
 [15]
Derek
J. S. Robinson, The vanishing of certain homology and cohomology
groups, J. Pure Appl. Algebra 7 (1976), no. 2,
145–167. MR 0404478
(53 #8280)
 [16]
Dock
Sang Rim, Modules over finite groups, Ann. of Math. (2)
69 (1959), 700–712. MR 0104721
(21 #3474)
 [17]
John
Stallings, Homology and central series of groups, J. Algebra
2 (1965), 170–181. MR 0175956
(31 #232)
 [18]
Richard
G. Swan, Periodic resolutions for finite groups, Ann. of Math.
(2) 72 (1960), 267–291. MR 0124895
(23 #A2205)
 [19]
Richard
G. Swan, Induced representations and projective modules, Ann.
of Math. (2) 71 (1960), 552–578. MR 0138688
(25 #2131)
 [20]
Richard
G. Swan, The Grothendieck ring of a finite group, Topology
2 (1963), 85–110. MR 0153722
(27 #3683)
 [21]
C.
T. C. Wall, Finiteness conditions for
𝐶𝑊complexes, Ann. of Math. (2) 81
(1965), 56–69. MR 0171284
(30 #1515)
 [1]
 K. S. Brown, Euler characteristics of discrete groups and spaces, Invent. Math. 27 (1974), 229264. MR 0385007 (52:5877)
 [2]
 K. S. Brown and P. K. Kahn, Homotopy dimension and simple cohomological dimension of spaces, Comment. Math. Helv. 52 (1977), 111127. MR 0438336 (55:11251)
 [3]
 G. Carlsson, A counterexample to a conjecture of Steenrod, Invent. Math. 64 (1981), 171174. MR 621775 (82j:57036)
 [4]
 W. Dwyer, Vanishing homology over nilpotent groups, Proc. Amer. Math. Soc. 49 (1975), 259261. MR 0374242 (51:10442)
 [5]
 K. Ehrlich, The obstruction to the finiteness of the total space of a fibration, Michigan Math. J. 28 (1981), 1938. MR 600412 (82i:55014)
 [6]
 K. Hoechsmann, P. Roquette and H. Zassenhaus, A cohomological characterization of finite nilpotent groups, Arch. Math. (Basel) 19 (1968), 225244. MR 0227278 (37:2863)
 [7]
 P. J. Hilton, Homotopy theory and duality, Gordon & Breach, New York, 1966.
 [8]
 R. H. Lewis, Equivariant cofibrations and nilpotency, Trans. Amer. Math. Soc. 267 (1981), 139155. MR 621979 (82j:55009)
 [9]
 , Ph. D. dissertation, Cornell University, 1977.
 [10]
 G. Mislin, Wall's obstruction for nilpotent spaces, Topology 14 (1975), 311317. MR 0394647 (52:15448)
 [11]
 , Finitely dominated nilpotent spaces, Ann. of Math. (2) 103 (1976), 547556. MR 0415607 (54:3690)
 [12]
 , Groups with cyclic Sylow subgroups..., Comment. Math. Helv. 52 (1977), 373391. MR 0448344 (56:6651)
 [13]
 , The geometric realization of Wall obstructions..., Math. Proc. Cambridge Philos. Soc. 87 (1980), 199206. MR 553576 (81f:55011)
 [14]
 G. Mislin and K. Varadarajan, The finiteness obstructions for nilpotent spaces lie in , Invent. Math. 53 (1979), 185191. MR 560413 (81e:55011)
 [15]
 D. J. S. Robinson, The vanishing of certain homology and cohomology groups, J. Pure Appl. Algebra 7 (1976), 145167. MR 0404478 (53:8280)
 [16]
 D. S. Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700712. MR 0104721 (21:3474)
 [17]
 J. Stallings, Homology and central series of groups, J. Algebra 2 (1965), 170181. MR 0175956 (31:232)
 [18]
 R. Swan, Periodic resolutions for finite groups, Ann. of Math. (2) 72 (1960), 267291. MR 0124895 (23:A2205)
 [19]
 , Induced representations and projective modules, Ann. of Math. (2) 71 (1960), 552578. MR 0138688 (25:2131)
 [20]
 , The Grothendieck ring of a finite group, Topology 2 (1963), 85110. MR 0153722 (27:3683)
 [21]
 C. T. C. Wall, Finiteness conditions for complexes, Ann. of Math. (2) 81 (1965), 5669. MR 0171284 (30:1515)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
55P99,
20C07,
57M99
Retrieve articles in all journals
with MSC:
55P99,
20C07,
57M99
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507928254
PII:
S 00029947(1985)07928254
Article copyright:
© Copyright 1985
American Mathematical Society
